turbulent boundary layer on a flat plate ( dp e / dx =0)

2004 Fluid Mechanics II Prof. António Sarmento - DEM/IST

Contents:– 1/7 velocity law;– Equations for the turbulent boundary layer with zero

pressure gradient (dpe/dx=0);– Virtual origin of the boundary layer;– Hydraulically smooth and fully rough flat plates.

Turbulent Boundary Layer on a flat plate (dpe/dx=0)


Boundary Layer Introdution Transition from laminar to turbulent regime:

Ux

v

forcesiscosity forces inertiaRe x – Distance to the leading edge

• Beginning of the BL 0x 0Re Laminar flow

•Sufficiently long plate : Re increases

Critical Re(5105)

Transition to turbulent

00 yyu very large

00 yyu decreases


Boundary Layer Introdution Turbulent regions of the BL:

– Linear sub-layer (no turbulence);– Transition layer;– Central region – logaritmic profile zone (turbulence not affected by the wall);– External zone (turbulent vortices mixed with non-turbulent outside flow).


Law of the wall

yuy

uuu

Experimental results from the law of the wall

0u


Law of the wall

Characteristics of the velocity profile u*=f(y*):

o Linear, laminar or viscous sub-layer5* y ** yu

o Central region30* y 5.5log75.5 *10

* yu

o Transition layer530 * y


Other approxmations for u=u(y)

o Take for any y5.5log75.5 *10

* yu

o Take - less reliable approximation, but easier to apply;

does not allow to calculate the shear stress in the wall.

7/1** 7.8 yu


Bases:–Von Kárman equation: mdx

dU 20

Note 1: the velocity profile in the BL follows the law of the wall , but this law has a less convenient form.

Note 2: as we saw in the laminar case, the integral parameters of the BL are little affected by the shape of the velocity profile

– Velocity profile (empirical): 71

7,8

yuuu

(Flat plates and ReL107)



Shear stress on the wall:

71

7,8

yuuu

412

0 0227,0

UU

Note: this expression relates 0 with (still unknown).

y71

7,8

uuU

0u

81

1506,0

UUu



=7/72

71

7,8

yuuu

y71

7,8

uuU

As we saw:

71

y

Uu

ydUu

Uu

m

1

0

1

a

am Conclusion:


7/72=0,0972<0,133 (Laminar BL)


On the other hand:71

y

Uu

ydUu

d

1

0

1

Form Factor:


81

d

29,1*

m

d Laminar BL => 2,59

The fuller the velocity profile is, closer to 1 the Form Factor is.


Note: xo is the point where =0. In general we choose xo to be in the beginning of the turbulent BL.

Von Kármàn Equation: mdxdU 2

0 dxdaU 2

Equation to 0:41

20 0227,0

UU

5441045

0 0284,0

Ua

xx



BL evolution on the flat plate:

xc x0

Laminar BL Turbulent BL

Transtion zone

cc Ux Re

(Rec5,5105) cc

c xRe5

0

c



Case 1 – the section of interest is very far away from the critical section (x>>xc): the BL is assumed to be turbulent from the beginning of the plate (x0=0=0).

5154

058,0

Uax

412

0 0227,0

UU

51

2 Re116,0

21

LD

a

LU

DC

251

0 Re0463,0 Ua

x

Valid if L>>xc (or ReL>>Rec). L is the plate lenght



Case 2 – the section of interest is not very faraway from the critical section: the transition zone is not considered

=> m0mc and x0=xc.

dxdc m

f2

From the Von Kármán equation

002 mcmtrans UD

mcm 0

cT

L

aa 0

am


aL=0,133 (Blasius)aT=7/72

dxxcUDx

xftrans

c

0

2

21


Virtual origin of the turbulent BL: xv

xv xc=xo


5154

058,0

Uax

4145

01,35

Uaxx cv

Would be as if the BL started turbulent from xv to reache 0 in x0.


Case 2: calculation of the drag on the plate.

TL DDD

dxxdxxc

vv

x

x

L

x 00

BlasiusDc CxU 2

21

51

2

Re116,0

21

vxLv

axLU


51

2

Re116,0

21

vc xxvc

axxU

xv xc=xo

2004 Fluid Mechanics II Teacher António Sarmento - DEM/IST

Correlations for higher Re:

58,210 Relog

455,0

LDC for Re109

LLDC

Re60

88,1Relog0776,0

210

for 3106 Re109



Hidraulically smooth plates if 5

us

167,0

024,0

LCD

All the contents studied before are for smooth plates

Turbulent Boundary Layer

Hidraulicaly fully rough plates if 80

us


Contents:– 1/7 Law of velocities;– Turbulent boundary layer expressions with dpe/dx null above

a flat plate;– Virtual origin of the boundary layer;– Hydraulically smooth and fully rough plates.



Sources:– Sabersky – Fluid Flow: 8.9– White – Fluid Mechanics: 7.4



A plate is 6 m long and 3 m wide and is immersed in a water flow (=1000 kg/m3, =1,1310-6 m2/s) with na undisturbed velocity of 6 m/s parallel to the plate. Rec=106. Compute:

a) The thickness of the BL at x=0,25 m; b) The thickness of the BL at x=1,9 m; c) The total drag on the plate; d) The maximum roughness on the plate for it to be hydraulically

smooth.

Exercise


L= 6 m; b=3 m; =1000 kg/m3; =1,1310-6 m2/s; U= 6 m/s; Rec=106.

a) Thickness of the BL at x1=0,25 m?

Exercise: solution

610Re Uxc

c m 188,00 xxc m 00094,0Re5 ccc x

m 00129,0727

133,00 c

T

Lc a

a

If we had addmited that the BL grew turbulent from the beginning:

54410145

01 0284,0

Ua

xx

m 0056,0058,05154

11

Uax

m 0027,0

In this case, the result would be significantly different



b) Thickness of the BL at x2=1,9 m?

Exercise: solution

610Re Uxc

c m 188,00 xxc m 00094,0Re5 ccc x

m 00129,0727

133,00 c

T

Lc a

a


54410245

02 0284,0

Ua

xx

m 0283,0058,05154

21

Uax

m 0264,0

In this case, the result would have a much smaller difference



c) Total drag on the plate?

Exercise: solution

baxxaxLxUDvcv xx

vcxL

vc

c

51

51

2

ReRe116,0

Re328,1

21

4145

01,35

Uaxx cv m 161,0For a 1/7 velocity law => a=7/72 =>

N 5,1452


N 2,1489Re2

1116,051

2

baLUD

L

Difference of 2,5%



c) Total drag on the plate?

Exercise: solution

L % dif.2 8,23 5,14 3,85 36 2,5

10 1,5100 0,16

Difference between computing D taking into account the laminar BL or assuming turbulent

from the leading edge.



d) Maximum roughness on the plate to be hidraulically smooth?

Exercise: solution

5

us

0u

41

0

20 0227,0

UU

It is necessary that: with

Where is 0 bigger? In the beginning of the turbulent BL

m/s 3,0

Pa 86,89

mm 0188,0

u5

turbulent boundary layer on a flat plate ( dp e / dx =0)

Documents

antnio sarmento demistlaw

antnio sarmento demiston

antnio sarmento demistas

antnio sarmento demistbases

antnio sarmento demistnote

antnio sarmento demistcase

uyfluid mechanics

layerfluid mechanics